a limit to height to which an aeroplane can ascend, and the
exact manner in which tins makes itself felt aerodynamically
becomes clear from the foregoing simple analysis of perform
ance. In the same manner it is proposed to consider how
the horse-power expended in turning
affects the performance. Suppose the
machine (Fig. 4) has reached the con.
ditions of steady horizontal flight
with velocity V in a circle of radius
R, the rudder and other controls
having been set to the appropriate
positions to maintain the steady
turning. The angle of bank being
<p the equations of motion become :—
Lsin <p = W/g. V«/R (1)
L cos <p = W (2
D - T (3)
terms of the horse-power required to drive the machine at
the equivalent velocity of horizontal flight
V, = V cos<p,
- V/(i + V'/^R2)1". (10)
no A.
V W
neglecting the relatively small components of the force on
the rudder and the difference in lift and drag on the two
wings due to the fact that the outer wing is moving more
rapidly than the inner. For a given speed and flight path
the requisite angle of bank is immediately given by
tan <p = V*/gR. It follows at once that to fly in a circle of
small radius with a given velocity the angle of bank must
be steep, but equation (2) indicates at the same time that in
order that the machine will not drop the lift must be imme
diately augmented, demanding an increased angle of attack
compared with horizontal flight. This involves a greater
expenditure of horse-power.
In general performance curves for any machine usually
give a measure of the horse-power required to maintain
horizontal flight at various speeds. From such a curve the
corresponding horse-power required to maintain circular
flight may easily be derived. In either case the horse-power
is expended in overcoming the drag. Comparing the two
cases, horizontal flight at velocity V and circular flight at
the same speed but necessarily at a different angle of attack,
H, - T,V - D,V - Ap(D,),V» (4)
for circular flight,
Hs - T2V = D3V - Ap(Dv)2V» (5)
for horizontal flight, where (Dr), and (D,-), are the appropriate
drag coefficients. Therefore
H, - H2(D,),/(D,)2. (6)
Now W = Ap(Lt),V2 cos <p from equation (2), and
W - Ap(Lt)2Va for horizontal flight. Therefore
(L<)./(L<-)i= i/cos<p. (7)
Suppose the machine flying in straight line flight with
velocity V, at the angle of attack corresponding to the lift
coefficient (Lr)„ that of the previous circular flight, then
W = Ap(L,)iV,»
and W - Ap(L,)3V«,
as before, where V is the horizontal flight equivalent, from
this point of view to the circular flight.
Therefore (L,),/(L,), - V'/V,'
hence V, — V N/cos <p (8)
If H', and H* be the horse-powers required to drive the
machine in horizontal flight at velocities V, and V respectively
then H., = DSV = Ap(D,)2V»,
and H'^D.V, = Ap(D,),V,»,
therefore (D,),/(D,), - H',/H,. Vs/V,»
-H',/H,. i/(cos<p)»'2.
This gives finally, using (6),
H, -HV(cos<p)»'2 (9)
expressing the horse-power required to maintain the machine
in horizontal circular flight, of radius R and velocity V, in
The method of obtaining the horse-power curve for circular
flight is at once obvious from these formulae. The horse
power for horizontal flight at velocity V ^/cos ^> is divided by
(cosip*'s) and plotted on a velocity base at abscissa V.
In Fig. 5 AD represents the horse-power available from
the propeller of a particular machine, and CB the curve
giving the horse-power required at various speeds for horizontal
flight. The curves of the type PQ representing the horse
power for circular flight at various radii and speeds are
obtained by the method indicated above.
Since SM represents the horse-power used up during the
flight, LS measures the surplus horse-power available, say,
for climb.
The curve of the type PSQ, passing through L, will, except
in the cases about to be mentioned, fix the minimum radius
of the circle at the velocity determined by M. If CN is the
locus of the points on each curve corresponding to maximum
.lift coefficient, none of the curves of the system can possibly
go beyond CN, and the points of intersection of this system
with this line determine the maximum radii of circular flight
at the corresponding speeds within the range limited by C
and N. The minimum possible radius corresponds to the
curve touching AB, in this case approximately 180 ft.
It must be remembered that in the type of flight con
templated it is necessary that the angle of attack of the
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machine should correspond to a higher lift coefficient than
that of horizontal flight at the same speed, in order to sustain
the weight when banked. The corresponding drag coeffi
cient would then be greater or less than that of horizontal
flight, according to whether the point considered lies above
or below the position of minimum drag, excluding those cases
'4